The field of the invention is that of the transmission or broadcasting of digital data, or of analog and sampled data, designed to be received in particular by moving bodies. More specifically, the invention relates to signals produced by means of new forms of modulation as well as to the corresponding techniques of modulation and demodulation.
For many years now, it has been sought to build modulation schemes adapted to highly non-stationary channels, such as channels for transmission towards moving bodies. In such channels, the signal sent out is affected by phenomena of fading and multiple paths. The work carried out by the CCETT within the framework of the European project EUREKA 147 (DAB: Digital Audio Broadcasting) has shown the value, for channels of this type, of multicarrier modulation and especially of OFDM (Orthogonal Frequency Division Multiplexing).
OFDM has been chosen within the framework of this European project as the basis of the DAB standard. This technique can also be chosen as a modulation technique for digital video broadcasting (DVB). However, a certain number of limitations (specified hereinafter) are observed when dealing with the problem of modulation encoded with high spectral efficiency such as the modulation required for digital television applications.
The invention can be applied in many fields, especially when high spectral efficiency is desired and when the channel is highly non-stationary.
A first category of applications relates to terrestrial digital radio-broadcasting, whether of images, sound and/or data. In particular, the invention can be applied to synchronous broadcasting which intrinsically generates long-term multiple paths. It can also advantageously be applied to broadcasting toward moving bodies.
Another category of applications relates to digital radiocommunications. The invention can be applied especially in systems of digital communications with moving bodies at high bit rates, in the framework for example of the UMTS. It can also be envisaged for high bit rate local radio networks (of the HIPERLAN type).
A third category of applications is that of underwater transmission. The transmission channel in underwater acoustics is highly disturbed because of the low speed of transmission of acoustic waves in water. This leads to a major spread of the multiple paths and of the Doppler spectrum. The techniques of multicarrier modulation, and especially the techniques that are an object of the present invention, are therefore well suited to this field.
Before presenting the signals according to the invention, a description is given here below of the known signals. This description is based on a general approach to multicarrier signals, defined by the inventors, that is novel per se. This general approach has indeed no equivalent in the prior art and is no way obvious to those skilled in the art. It must therefore be considered to be a part of the invention and not as forming part of the prior art.
The signals of interest are real signals (an electrical magnitude for example), that have finite energy and are a function of time. The signals may therefore be represented by real functions of L2(R). Furthermore, these signals are limited band signals and their spectrum is contained in       [                            f          c                -                  w          2                    ,                        f          c                +                  w          2                      ]    ,
fc being the xe2x80x9ccarrier frequencyxe2x80x9d of the signal. It is therefore possible, in an equivalent manner, to represent a real signal a(t) by its complex envelope s(t) with:
s(t)=exe2x88x92ixcfx80fctFA[a](t)xe2x80x83xe2x80x83(1)
where FA designates the analytical filter.
The signal s(t) belongs to a vector subspace   (            characterized  by  the  limitation  of  the  band  to        ±          w      2        )
of the space of the complex functions of a real variable with a summable square L2(R). This vector space can be defined in two different ways, depending on whether the building is done on the field of the complex values or on the field of the real values. With each of these spaces, it is possible to associate a scalar product with a value in C or in R and to build a Hilbert space. H designates the Hilbert space built on the field of the complex values and HR designates the Hilbert space built on the field of the real values.
The corresponding scalar values are written as follows:                                                         ⟨                              x                |                y                            ⟩                        =                                          ∫                R                            ⁢                                                x                  ⁡                                      (                    t                    )                                                  ⁢                                                      y                    +                                    ⁡                                      (                    t                    )                                                  ⁢                                  xe2x80x83                                ⁢                                  ⅆ                  t                                ⁢                                  xe2x80x83                                ⁢                                  in  the  case  of  H                                                              ⁢                      
                    ⁢                      and                          ⁢                  
                                    (        2        )                                                      ⟨                          x              |              y                        ⟩                    R                =                  ℛe          ⁢                                    ∫              R                        ⁢                                          x                ⁡                                  (                  t                  )                                            ⁢                                                y                  +                                ⁡                                  (                  t                  )                                            ⁢                              xe2x80x83                            ⁢                              ⅆ                t                            ⁢                              xe2x80x83                            ⁢                              in  the  case  of                              ⁢                              H                R                                                                        (        3        )            
The associated standards are obviously identical in both cases:                               "LeftDoubleBracketingBar"          x          "RightDoubleBracketingBar"                =                              [                                          ∫                R                            ⁢                                                                    "LeftBracketingBar"                                          x                      ⁡                                              (                        t                        )                                                              "RightBracketingBar"                                    2                                ⁢                                  ⅆ                  t                                                      ]                                1            /            2                                              (        4        )            
The general principles of the OFDM are presented for example in the French patent FR-86 09622 filed on Jul. 2nd, 1986. The basic idea of the technique is that of transmitting encoded signals as coefficients of elementary waveforms that are confined as far as possible in the time-frequency plane and for which the transmission channel may be considered to be locally stationary. The channel then appears to be a simple multiplier channel characterized by the distribution of the modulus of the coefficients, which follows a law of Rice or of Rayleigh.
Protection is then provided against fading phenomena by means of a code. This code can be used in weighted decision mode in association with time and frequency interlacing, which ensures that the signals playing a part in the minimum meshing of the code are affected, to the utmost possible extent, by independent fading phenomena.
This technique of encoding with interlacing in the time-frequency plane is known as COFDM. It is described for example in the document [22] (see Appendix 1 (to simplify the reading, most of the prior art references are listed in Appendix 1. This Appendix as well as Appendices 2 and 3 must of course be considered to be integral parts of the present description)).
There are essentially two types of known OFDM modulations. The terms applied in the literature are often ambiguous. Here we introduce new appellations that are more precise while recalling their relation with the existing literature. We shall use the generic name OFDM followed by a suffix specifying the type of modulation within this group.
2.3.1 Theoretical Principles
A first category of modulation is constituted by a multiplex of QAM (Quadrature Amplitude Modulation) carriers or possibly QPSK (Quadrature Phase Shift Keying) carriers in the particular case of binary data elements. Hereinafter, this system shall be called OFDM/QAM. The carriers are all synchronized and the carrier frequencies are spaced out in reverse to the symbol time. Although the spectra of these carriers overlap, the synchronization of the system makes it possible to ensure orthogonality between the symbols sent out by the different carriers.
The references [1] to [7] give a good idea of the literature available on this subject.
For greater simplicity in the writing, and according to the novel approach of the invention, the signals will be represented by their complex envelope described here above. Under these conditions, the general equation of an OFDM/QAM signal is written as follows:                               s          ⁡                      (            t            )                          =                              ∑                          m              ,              n                                ⁢                                    a                              m                ,                n                                      ⁢                                          x                                  m                  ,                  n                                            ⁡                              (                t                )                                                                        (        5        )            
The coefficients am,n take complex values representing the data sent. The functions xm,n(t) are translated into the time-frequency space of one and the same prototype function x(t):                               xe2x80x83                ⁢                              x            ⁡                          (              t              )                                =                      {                                                                                1                                                                  τ                        0                                                                                                                                                        si                      ⁢                                              "LeftBracketingBar"                        t                        "RightBracketingBar"                                                              ≤                                                                  τ                        0                                            /                      2                                                                                                                    0                                                  elsewhere                                                                                        (        6        )            
xm,n(t)=e2ixcfx80mxcexd0tx(txe2x88x92nxcfx840) with xcexd0xcfx840=1xe2x80x83xe2x80x83(7)
xcfx86 being any phase that can be arbitrarily set at 0. The function x(t) is centered, namely its first order moments are zero, giving:
∫t|x(t)|2dt=∫f|X(f)2df=0,xe2x80x83xe2x80x83(8)
X(f) designating the Fourier transform of x(t).
Under these conditions, it is observed that:
∫t∥xm,n(t)∥2dt=nxcfx840
∫t∥Xm,n(f)∥2dt=nxcexd0xe2x80x83xe2x80x83(9)
The barycenters of the basic functions therefore form a lattice of the time-frequency plane generated by the vectors (xcfx840,0) and (0,xcexd0), as shown in FIG. 1. This lattice has a density of one, namely xcexd0xcfx840=1. Reference may be made to the article [9] for a more detailed discussion on this subject.
The prototype function x(t) has the special characteristic wherein the functions {xm,n} are mutually orthogonal and more specifically constitute a Hilbert base of L2(R), giving:                               ⟨                                    x                              m                ,                n                                      |                          x                                                m                  xe2x80x2                                ,                                  n                  xe2x80x2                                                              ⟩                =                  {                                                                                          1                    ⁢                                          xe2x80x83                                        ⁢                    if                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              m                        ,                        n                                            )                                                        =                                      (                                                                  m                        xe2x80x2                                            ,                                              n                        xe2x80x2                                                              )                                                                                                                        0                  ⁢                                      xe2x80x83                                    ⁢                  if                  ⁢                                      xe2x80x83                                    ⁢                  not                                                                                        (        10        )            
Projecting a signal on this basis is equivalent simply to breaking down the signal into sequences with a duration of xcfx840 and representing each of these sequences by the corresponding Fourier series development. This type of breakdown is a first step towards a localization both in time and in frequency as opposed to the standard Fourier analysis which provides for perfect frequency localization with a total loss of temporal information.
Unfortunately, while the temporal localization is excellent, the frequency localization is far less efficient owing to the slow decreasing function of X(f). The Balian-Low-Coifman-Semmes theorem (see [9], p. 976) furthermore shows that if X designates the Fourier transform of x, then tx(t) and fX(f) cannot simultaneously be summable squares.
2.3.2 The OFDM/OAM with Guard Interval
Generally, the tolerance of an OFDM modulation with respect to multiple paths and Doppler spreading can be characterized by a parameter that comprehensively measures the variation of the level of inter-symbol interference (ISI) as a function of a temporal or frequency shift. The justification of this concept is given in Appendix 2. This tolerance parameter is called "xgr" and is defined by the relationship:
"xgr"=1/4xcfx80xcex94txcex94fxe2x80x83xe2x80x83(11)
with:
xcex94t2∫|x(t)|2dt=∫t|x(t)|2dtxe2x80x83xe2x80x83(12)
xcex94f2∫|X(f)|2dt=∫f|X(f)|2dtxe2x80x83xe2x80x83(13)
By virtue of Heisenberg""s inequality, "xgr" cannot exceed unity.
Given the Balian-Low-Coifman-Semmes theorem referred to here above, the parameter "xgr" is equal to 0 for the OFDM/QAM. This is a major defect of the OFDM/QAM modulation as described here above. This is characterized in practice by high sensitivity to temporal errors and consequently multiple paths.
This defect can be circumvented by the use of a guard interval described for example in [5]. This is a device consisting in extending the rectangular window of the prototype function. The density of the lattice of base symbols is then strictly smaller than unity.
This technique is possible because an infinity of translated versions of the initial symbol is found within a symbol extended by a guard interval. Of course, this works only because the prototype function is a rectangular window. In this sense, the OFDM/QAM with a guard interval is a unique and singular point.
OFDM/QAM modulation with guard interval is the basis of the DAB system. This guard interval makes it possible to limit inter-symbol interference at the cost of a loss of performance since a part of the information transmitted is not really used by the receiver but is used only to absorb the multiple paths.
Thus, in the case of the DAB system, where the guard interval represents 25% of the useful symbol, the loss is 1 dB. Furthermore, there is an additional loss due to the fact that to obtain a given comprehensive spectral efficiency, it is necessary to compensate for the loss due to the guard interval by a greater efficiency of the code used.
This loss is marginal in the case of the DAB system because the spectral efficiency is low. On the contrary, if it is sought to obtain an overall spectral efficiency of 4 bits/Hz, it is necessary to use a 5 bit/Hz code giving, according to the Shannon theorem, a loss of the order of 3 dB. The total loss is therefore in this case about 4 dB.
2.3.3 Other OFDM/QAM Systems
It is possible to conceive of other systems of the OFDM/QAM type. Unfortunately, no filtered QAM modulation, namely one using a conventional half-Nyquist (or more specifically xe2x80x9cNyquist square rootxe2x80x9d) type of shaping verifies the requisite constraints of orthogonality. The known prototype functions verifying the requisite criteria of orthogonality are:
the rectangular window;
the cardinal sine.
These two examples are trivial and appear to be dual with respect to each other by the Fourier transform. The case of the rectangular window corresponds to the OFDM/QAM without guard interval. The case of the cardinal sine corresponds to a standard frequency multiplex (namely one where the carriers have disjoint spectra) with a 0% roll-off which is an asymptotic case that is difficult to achieve in practice.
In each of these cases, it is observed that the prototype function is perfectly limited either in time or in frequency but has a mediocre decrease (in 1/t or 1/f) in the dual domain.
The Balian-Low-Coifman-Semmes theorem furthermore leaves little hope that there might exist satisfactory solutions. As indicated here above, this theorem shows that tx(t) and fX(f) cannot simultaneously have a summable square. They can therefore be no hope of finding a function x(t) such that x(t) and X(f) decrease simultaneously with an exponent smaller than xe2x88x923/2.
This furthermore does not rule out the possibility that there exist functions that are satisfactory from the viewpoint of an engineer. However, a recent article [10] dealing with this subject shows another exemplary prototype function having the requisite properties. The shape of the prototype function proposed in this article is very far from what may be hoped for in terms of temporal concentration. It is therefore probable that there is no satisfactory OFDM/QAM type solution.
In conclusion, the OFDM/QAM approach corresponding to the use of a lattice with a density 1 and complex coefficients am,n can be put into practice only in the case of a rectangular temporal window and in the case of the use of a guard interval. Those skilled in the art seeking other modulations will therefore have to turn towards the techniques described here below under the designation of OFDM/OQAM.
A second category of modulations uses a multiplex of OQAM (Offset Quadrature Amplitude Modulation) modulated carriers. Hereinafter, this system shall be called OFDM/OQAM. The carriers are all synchronized and the carrier frequencies are spaced out by half of the reverse of the symbol time. Although the spectra of these carriers overlap, the synchronization of the system and the choices of the phases of the carriers can be used to guarantee the orthogonality between the symbols put out by the different carriers. The references [11-18] give a clear picture of the literature available on this subject.
For greater simplicity in the writing, the signals are represented in their analytical form. Under these conditions, the general equation of an OFDM/OQAM signal can be written as follows:                               s          ⁡                      (            t            )                          =                              ∑                          m              ,              n                                ⁢                                    a                              m                ,                n                                      ⁢                                          x                                  m                  ,                  n                                            ⁡                              (                t                )                                                                        (        14        )            
The coefficients am,n assume real values representing the data elements transmitted. The functions xm,n(t) are translated in the time-frequency space of one and the same prototype function x(t):                     {                                                                                                                        x                                              m                        ,                        n                                                              ⁡                                          (                      t                      )                                                        =                                                            ⅇ                                              ⅈ                        ⁡                                                  (                                                                                    2                              ⁢                                                              xe2x80x83                                                            ⁢                              π                              ⁢                                                              xe2x80x83                                                            ⁢                              m                              ⁢                                                              xe2x80x83                                                            ⁢                                                              ν                                0                                                            ⁢                              t                                                        +                            ϕ                                                    )                                                                                      ⁢                                          x                      ⁡                                              (                                                  t                          -                                                      n                            ⁢                                                          xe2x80x83                                                        ⁢                                                          τ                              0                                                                                                      )                                                                                            ⁢                                  xe2x80x83                                                                                                      if                  ⁢                                      xe2x80x83                                    ⁢                  m                                +                                  n                  ⁢                                      xe2x80x83                                    ⁢                  est                  ⁢                                      xe2x80x83                                    ⁢                  pair                  ⁢                                      xe2x80x83                                    ⁢                  is                  ⁢                                      xe2x80x83                                    ⁢                  even                                                                                                                                              x                                          m                      ,                      n                                                        ⁡                                      (                    t                    )                                                  =                                                      ⅈⅇ                                          ⅈ                      ⁡                                              (                                                                              2                            ⁢                                                          xe2x80x83                                                        ⁢                            π                            ⁢                                                          xe2x80x83                                                        ⁢                            m                            ⁢                                                          xe2x80x83                                                        ⁢                                                          ν                              0                                                        ⁢                            t                                                    +                          ϕ                                                )                                                                              ⁢                                      x                    ⁡                                          (                                              t                        -                                                  n                          ⁢                                                      xe2x80x83                                                    ⁢                                                      τ                            0                                                                                              )                                                                                                                                            if                  ⁢                                      xe2x80x83                                    ⁢                  m                                +                                  n                  ⁢                                      xe2x80x83                                    ⁢                  is                  ⁢                                      xe2x80x83                                    ⁢                  odd                                                                    ⁢                  xe2x80x83                                    (        15        )            
with xcexd0xcfx840=1/2.
xcfx86 being any phase that can be arbitrarily set at 0.
The barycenters of the basic functions therefore form a lattice of the time-frequency plane generated by the vectors (xcfx840, 0) and (0, xcexd0), as shown in FIG. 2.
This lattice has a density 2. The functions xm,n(t) are mutually orthogonal in the sense of the scalar product in R. In the known approaches, the prototype function is limited in frequency in such a way that the spectrum of each carrier overlaps only that of the adjacent carriers. In practice, the prototype functions considered are even-parity functions (real or possibly complex) verifying the following relationship:                     {                                                                              X                  ⁡                                      (                    f                    )                                                  =                0                                                                                      if                  ⁢                                      xe2x80x83                                    ⁢                                      "LeftBracketingBar"                    f                    "RightBracketingBar"                                                  ≥                                  v                  0                                                                                                                                                                    "LeftBracketingBar"                                              X                        ⁡                                                  (                          f                          )                                                                    "RightBracketingBar"                                        2                                    +                                                            "LeftBracketingBar"                                              X                        ⁡                                                  (                                                      f                            -                                                          ν                              0                                                                                )                                                                    "RightBracketingBar"                                        2                                                  =                                  1                  /                                      ν                    0                                                                                                                        if                  ⁢                                      xe2x80x83                                    ⁢                  0                                ≤                f                ≤                                  ν                  0                                                                                        (        16        )            
A possible choice for x(t) is the pulse response of a half-Nyquist filter with 100% roll-off, namely:                               X          ⁡                      (            f            )                          =                  {                                                                                          1                                                                  ν                        0                                                                              ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                  π                  ⁢                                      xe2x80x83                                    ⁢                  f                  ⁢                                      xe2x80x83                                    ⁢                                      τ                    0                                                                                                                    if                    ⁢                                          xe2x80x83                                        ⁢                                          "LeftBracketingBar"                      f                      "RightBracketingBar"                                                        ≤                                      v                    0                                                                                                      0                                            elsewhere                                                                        (        17        )            
When x(t) and its Fourier transform are observed, it is noted that X(f) has a bounded support and that x(t) decreases in txe2x88x922, i.e. this is a result substantially better than the theoretical limit resulting from the Balian-Low-Coifman-Semmes theorem. The elementary waveforms are better localized in the time-frequency plane than in the case of the OFDM/QAM, which gives this modulation a better behavior in the presence of multiple paths and of Doppler phenomena. As above, it is possible to define the parameter "xgr" measuring the tolerance of the modulation to the delay and to the Doppler phenomenon. This parameter "xgr" is equal to 0.865.
Another approach, described in the French patent application FR-95 05455 filed on May 2, 1995 (as yet unpublished) on behalf of the Applicants filing the present Application, consists in using a different prototype function. This prototype function x(t) is an even-parity function, zero outside the interval [xe2x88x92xcfx840,xcfx840], and verifies the relationship:   {      "AutoLeftMatch"                                                      x              ⁡                              (                t                )                                      =            0                                                              if              ⁢                              xe2x80x83                            ⁢                              "LeftBracketingBar"                t                "RightBracketingBar"                                      ≥                          τ              0                                                                                                      "LeftBracketingBar"                                  x                  ⁡                                      (                    t                    )                                                  "RightBracketingBar"                            2                        =                                                            "LeftBracketingBar"                                      x                    ⁡                                          (                                              t                        -                                                  τ                          0                                                                    )                                                        "RightBracketingBar"                                2                            =                              1                /                                  τ                  0                                                                                                        if              ⁢                              xe2x80x83                            ⁢              0                        ≤            t             less than                           τ              0                                          
Advantageously, said prototype function x(t) is defined by:       x    ⁡          (      t      )        =      {                                                      1                                                τ                  0                                                      ⁢            cos            ⁢                          xe2x80x83                        ⁢            π            ⁢                          xe2x80x83                        ⁢                          t              /              2                        ⁢                          τ              0                                                                          if              ⁢                              xe2x80x83                            ⁢                              "LeftBracketingBar"                t                "RightBracketingBar"                                      ≤                          τ              0                                                            0                          elsewhere                    
This function may be considered to be dual by Fourier transform of the prototype function used in the case of the OFDM/OQAM. This particular case is called OFDM/MSK. The performance characteristics in terms of resistance to Doppler phenomena and multiple paths are equivalent to the OFDM/OQAM, and the making of the receiver is simplified.
Another approach, also described in the patent application FR-95 05455 referred to here above, consists of the use of a prototype function that is limited neither in time nor in frequency, but possesses properties of fast decreasing in the temporal as well as frequency domains. This prototype function x(t) is characterized by the equation:       x    ⁢          (      t      )        =            y      ⁢              (        t        )                                      τ          0                ⁢                              ∑            k                    ⁢                                    "LeftBracketingBar"                              y                ⁢                                  (                                      t                    -                                          k                      ⁢                                              xe2x80x83                                            ⁢                                              τ                        0                                                                              )                                            "RightBracketingBar"                        2                              
the function y(t) being defined by its Fourier transform Y(f):       Y    ⁢          (      f      )        =            G      ⁢              (        f        )                                      ν          0                ⁢                              ∑            k                    ⁢                                    "LeftBracketingBar"                              G                ⁢                                  (                                      f                    -                                          k                      ⁢                                              xe2x80x83                                            ⁢                                              ν                        0                                                                              )                                            "RightBracketingBar"                        2                              
where G(f) is a normalized Gaussian function of the type: G(f)=(2xcex1)1/4exe2x88x92xcfx80xcex1f2. In the case of the OFDM/IOTA modulation, the parameter a is set at 1. In this case, the corresponding prototype function, referenced ℑ, is identical to its Fourier transform.
The making of the receiver is simpler than in the case of the OFDM/OQAM, although it is slightly more complex than in the preceding case, but the performance is substantially superior.
These prior art systems have many drawbacks and limits, especially in very disturbed channels and when high efficiency is required.
The main problem of the OFDM/QAM system is that it imperatively requires the use of a guard interval. As indicated here above, this gives rise to a substantial loss of efficiency when high spectral efficiency values are aimed at.
Furthermore, the signals sent out are poorly concentrated in the frequency domain, which also limits the performance characteristics in the highly non-stationary channels. In particular, this spread makes it difficult to use equalizers.
Conversely, the frequency performance characteristics of the OFDM/OQAM are rather satisfactory and the problem of the loss related to the guard interval does not arise. By contrast, the pulse response of the prototype function has a relatively slow temporal decrease, namely a decrease in 1/x2.
This implies two types of difficulties. First of all, it may be difficult to truncate the waveform in a short interval of time. This implies complex processing in the receiver. Furthermore, this also implies possible systems of equalization.
In other words, the efficiency of the OFDM/OQAM techniques is greater than that of the OFDM/QAM techniques, but these techniques prove to be more complicated to implement and therefore costly, especially in receivers.
OFDM/MSK modulation has performance characteristics in terms of resistance to Doppler phenomena and multiple paths in relation to the OFDM/OQAM. These performance characteristics are below those of the OFDM/IOTA. By contrast, the temporal function of the prototype function simplifies the receiver.
OFDM/IOTA modulation has optimum performance characteristics in terms of resistance to Doppler phenomena and multiple paths. By contrast, the making of the receiver is more complex than in the case of the OFDM/MSK.
It is a goal of the invention in particular to overcome these different drawbacks and limitations of the prior art.
Thus, a goal of the invention is to provide a digital signal designed to be transmitted or broadcast to receivers, that can be used to obtain performance characteristics equivalent to those of the best known solution, namely the OFDM/IOTA solution, while at the same time improving the temporal response concentration, in particular so as to simplify the processing at the receiver.
The invention is also aimed at providing a signal of this kind enabling the making of receivers with limited complexity and cost, especially as regards demodulation and equalization.
An additional goal of the invention is to provide transmitters, receivers, methods of transmission or broadcasting, methods of reception and methods for the building, namely the definition, of a modulation corresponding to such a signal.
These aims as well as others that shall appear hereinafter are achieved according to the invention by a multicarrier signal designed to be transmitted to digital receivers, especially in a non-stationary transmission channel, corresponding to the frequency multiplexing of several elementary carriers each corresponding to a series of symbols, built on a non-orthogonal time-frequency lattice with a density 2.
Advantageously, this lattice is a quincunxial lattice in which, the spacing between two neighboring carriers being equal to xcexd0,
the symbol time xcfx840 is equal to a quarter of the inverse of the spacing xcexd0 between two neighboring carriers,
the symbols sent out on one and the same carrier are spaced out by two symbol times xcfx840.
the symbols sent out on adjacent carriers are offset by the symbol time xcfx840.
Preferably, each carrier undergoes a filtering operation for the shaping of its spectrum.
This filtering is chosen so that each symbol element is concentrated as far as possible both in the temporal field and the frequency field.
In particular, a signal of this kind may meet the following equation:       s    ⁡          (      t      )        =            ∑              m        +                  n          ⁢                      xe2x80x83                    ⁢          even                      ⁢                  a                  m          ,          n                    ⁢                        x                      m            ,            n                          ⁡                  (          t          )                    
where:
am,n is a real coefficient representing the signal source chosen in a predetermined alphabet of modulation;
m is an integer representing the frequency dimension;
n is an integer representing the temporal dimension;
t represents time;
xm,n(t) is a basic function translated into the time-frequency space of one and the same even-parity prototype function x(t) taking real or complex values, namely:
xm,n(t)=eixcfx86m,nei(2xcfx80m"ugr"0xcfx84+xcfx86)x(txe2x88x92nxcfx840) with "ugr"0xcfx840=1/2
with xcfx86m,n=(m+n+mn+(n2xe2x88x92m2)/2)xcfx80/2
where xcfx86 is an arbitrary phase parameter,
and where said basic functions {xm,n} are mutually orthogonal, the real part of the scalar product of two different basic functions being zero.
Thus, the invention is based on a system of modulation using prototype functions that are as concentrated as possible in the time-frequency plane. The value of this approach is that it makes available a modulation having performance characteristics identical to those of the OFDM/IOTA modulation while at the same time providing the benefit of a faster decrease pulse response.
In other words, an object of the invention relates to novel systems of modulation built, like the OFDM/IOTA modulation, on a lattice with a density 2. The essential difference with respect to systems that are already known is that the basic lattice is a quincunxial lattice with a density of 1/2.
Among the types of modulation proposed, there are modulations using prototype functions that are bounded neither in time nor in frequency but, on the contrary, have properties of fast decrease both in time and in frequency and an almost optimum concentration in the time-frequency plane.
Signals of this kind are in no way obvious to those skilled in the art, in view of the prior art. As indicated here above, there are basically two modes of building OFDM type modulations.
The first known building mode uses an orthogonal lattice with a density 1. This first approach uses a base for the breakdown of the signals where every signal is subdivided into intervals, each interval being then broken down in the form of Fourier series. This is the OFDM/QAM approach. The literature gives few examples of alternative approaches built on the same lattice, and the results obtained are of little practical interest [10].
Furthermore, the OFDM/QAM technique is the only one that can benefit from the method of the guard interval. The OFDM/QAM approach is therefore a singular feature that permits no extension.
The second known building mode uses an orthogonal lattice with the density 2. It combines a set of modulations such as the OFDM/OQAM, OFDM/MSK and OFDM/IOTA modulations. These modulations differ in the choice of the prototype function which is either frequency-bounded (OFDM/OQAM) or time-bounded (OFDM/MSK) or is bounded neither in time nor in frequency but has properties of fast decrease in both dimensions (OFDM/IOTA).
Consequently, it is not a simple matter to build new types of modulation that are not built on such orthogonal lattices.
All the variants of the invention described here below have the advantage of using a fast decrease prototype function in such a way that the function can be easily truncated.
The basic principle consists in building a prototype function that has the desired properties of orthogonality on a quincunxial type of lattice. The building method, which is described in detail in appendix 2, consists in effecting a 45xc2x0 rotation in the time-frequency plane on the basis of a prototype function having the desired properties of orthogonality on an orthogonal lattice. The operator that enables this rotation is not a classical operator. It may be likened to the square root of a Fourier transform and is denoted F1/2. The mathematical justification of this notation is given in appendix 3.
In principle, the method may be applied to any prototype function that enables the building of a Hilbert base on an orthogonal lattice with a density 1/2. On this basis, the prototype functions of the OFDM/OQA/ and of the OFDM/MSK are usable. They lead however to results that are complex functions. Consequently, the results obtained are of little practical value.
Let us consider the conditions needed for this building method to lead to a real function. Let us take an even-parity real prototype function x(t) enabling the generation of a Hilbert base on an orthogonal lattice with a density 2, and the function y(t) defined by:
y=F1/2x
where F1/2 is the square root operator of the Fourier transform as defined here above. At the level of the ambiguity functions (as defined in appendix 2), this operator carries out a rotation by an angle xe2x88x92xcfx80/4 in the time-frequency plane. In order that the function y(t) may be an even-parity real function, its ambiguity function should have a symmetry with respect to the time and frequency axes. This therefore means that the function x(t), in addition to the symmetry with respect to the time and frequency axes, has a symmetry with respect to the diagonals of the time-frequency plane. Such a property can be verified only if the function x(t) is identical to its Fourier transform. Now, at present, we know only one function having this property. This is the prototype function of the OFDM/IOTA, namely ℑ.
The function IOTA xcfx80/4 is therefore defined by the relationship:
ℑxcfx80/4=F1/2ℑ
This function is, by construction, a real, even-parity function. The ambiguity function of this function gets cancelled therefore on a quincunxial lattice.
The set of functions {ℑm,nxcfx80/4}defined by:
ℑm,nxcfx80/4(t)=eixcfx86m,neixcfx80(mxe2x88x92n)tℑxcfx80/4(txe2x88x92(m+n)/2)
with xcfx86m,n=(m+n+mn+(n2xe2x88x92m2)/2)xcfx80/2
constitute a Hilbert base.
In redefining the indices, the definition of this set may be rewritten as follows:
ℑm,nxcfx80/4(t)=eixcfx86m,neixcfx80mtℑxcfx80/4(txe2x88x92n/2), m+n even parity
with xcfx86m,n=(nxe2x88x92mn/2+(n2xe2x88x92m2)/4)xcfx80/2
The invention also relates to a method for the transmission of a digital signal especially in a non-stationary transmission channel, comprising the following steps:
the channel encoding of a digital signal to be transmitted, delivering real digital coefficients amen chosen out of a predetermined alphabet;
the building of a signal s(t) meeting the equation defined here above;
the transmission of a signal, having said signal s(t) as its complex envelope, to at least one receiver.
Advantageously, a method of this kind furthermore comprises a step of frequency and/or time interlacing applied to the binary elements forming said digital signal to be transmitted or to the digital coefficients am,n.
This makes it possible to provide for optimal performance characteristics in the non-stationary channels.
The invention also relates to the transmitters of a signal of this kind.
The invention also relates to a method for the reception of a signal as described here above, comprising the following steps:
the reception of a signal having, as its complex envelope, a signal r(t) corresponding to the signal s(t) at transmission;
the estimation of the response of the transmission channel comprising an estimation of the phase response xcex8m,n and of the amplitude response xcfx81m,n;
the demodulation of said signal r(t) comprising the following steps:
the multiplication of said signal r(t) by the prototype function x(t);
the aliasing of the filtered waveform modulo 2xcfx840;
the application of a Fourier transform (FFT);
the selection of the samples for which m+n is an even-parity value;
the correction of the phase xcex8m,n induced by the transmission channel;
the correction of the phase corresponding to the term eixcfx86m,n 
the selection of the real part of the coefficient obtained xc3xa3m,n corresponding to the transmitted coefficient amen weighted by the amplitude response xcfx81m,n of the transmission channel.
Preferably, this reception method comprises a step for the frequency and/or time de-interlacing of said real digital coefficients xc3xa3m,n and, possibly, of the corresponding values xcfx81m,n of the amplitude response of the channel, said de-interlacing being the reverse of an interlacing implemented at transmission and/or a step of weighted-decision decoding that is adapted to the channel encoding implemented at transmission.
The invention also relates to the corresponding receivers.
Finally, the invention also relates to a preferred method for the building of a prototype function x(t) for a signal as described here above. This method is presented in the appendices here below.